Acta Biotheoretica

, Volume 48, Issue 1, pp 47–63 | Cite as

Pattern Formation in a Nonlinear Membrane Model for Epithelial Morphogenesis

  • Larry A. Taber


A theoretical model is presented for pattern formation in an epithelium. The epithelial model consists of a thin, incompressible, viscoelastic membrane on an elastic foundation (substrate), with the component cells assumed to have active contractile properties similar to those of smooth muscle. The analysis includes the effects of large strains and material nonlinearity, and the governing equations were solved using finite differences. Deformation patterns form when the cells activate while lying on the descending limb of their total (active + passive) stress-stretch curve. Various one-dimensional and two-dimensional simulations illustrate the effects of spatial and temporal variations in passive stiffness, as well as the effects of foundation stiffness and stretch activation. The model can be used to examine the mechanical aspects of pattern formation in morphogenetic processes such as angiogenesis and myocardial trabeculation.


Pattern Formation Elastic Foundation Active Contractile Material Nonlinearity Deformation Pattern 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Brodland, G.W. and D.A. Clausi (1994). Embryonic Tissue Morphogenesis Modelled by FEM. Journal of Biomechanical Engineering 116: 146–155.Google Scholar
  2. Burnside, B. (1973). Microtubules and Microfilaments in Amphibian Neurulation American Zoology 13: 989–1006.Google Scholar
  3. Celia, M.A. and W.G. Gray (1992). Numerical Methods for Differential Equations, Englewood Cliffs, N.J.: Prentice-Hall.Google Scholar
  4. Eringen, A.C. (1962). Nonlinear Theory of Continuous Media, New York: McGraw-Hill.Google Scholar
  5. Ettensohn, C.A. (1985). Mechanisms of Epithelial Invagination. Quarterly Review of Biology 60: 289–307.Google Scholar
  6. Folkman, J. and C. Haudenschild (1980). Angiogenesis In Vitro. Nature 288: 551–556.Google Scholar
  7. Forgacs, G., R.A. Foty, Y. Shafrir and M.S. Steinberg (1998). Viscoelastic Properties of Living Embryonic Tissues: a Quantitative Study. Biophysical Journal 74: 2227–2234.Google Scholar
  8. Fristrom, D. (1988). The Cellular Basis of Epithelial Morphogenesis. A Review. Tissue & Cell 20: 645–690.Google Scholar
  9. Fung, Y.C. (1997). Biodynamics: Circulation, Second Ed., New York: Springer.Google Scholar
  10. Gordon, R. and G.W. Brodland (1987). The Cytoskeletal Mechanics of Brain Morphogenesis. Cell State Splitters Cause Primary Neural Induction. Cell Biophysics 11: 177–238.Google Scholar
  11. Hiruma, T. and R. Hirakow (1985). An Ultrastructural Topographical Study on Myofibrillogenesis in the Heart of the Chick Embryo during Pulsation Onset Period. Anatomy and Embryology 172: 325–329.Google Scholar
  12. Manasek, F.J. (1970). Histogenesis of the Embryonic Myocardium. American Journal of Cardiology 25: 149–168.Google Scholar
  13. Manoussaki, D., S.R. Lubkin, R.B. Vernon and J.D. Murray (1996). A Mechanical Model for the Formation of Vascular Networks In Vitro. Acta Biotheoretica 44: 271–282.Google Scholar
  14. Miller, C.E., M.A. Vanni, L.A. Taber and B.B. Keller (1997). Passive Stress-Strain Measurements in the Stage-16 and Stage-18 Embryonic Chick Heart. Journal of Biomechanical Engineering 119: 445–451.Google Scholar
  15. Murray, J.D. (1993). Mathematical Biology, 2nd Ed., New York: Springer-Verlag.Google Scholar
  16. Nagorcka, B.N., V.S. Manoranjan and J.D. Murray (1987). Complex Spatial Patterns from Tissue Interactions — an Illustrative Model. Journal of Theoretical Biology 128: 359–374.Google Scholar
  17. Odell, G.M., G. Oster, P. Alberch and B. Burnside (1981). The Mechanical Basis of Morphogenesis. I. Epithelial Folding and Invagination. Developmental Biology 85: 446–462.Google Scholar
  18. Oster, G. and P. Alberch (1982). Evolution and Bifurcation of Developmental Programs. Evolution 36: 444–459.Google Scholar
  19. Oster, G.F., J.D. Murray and A.K. Harris (1983). Mechanical Aspects of Mesenchymal Morphogenesis. Journal of Embryology and Experimental Morphology 78: 83–125.Google Scholar
  20. Taber, L.A., B.B. Keller and E.B. Clark (1992). Cardiac Mechanics in the Stage-16 Chick Embryo. Journal of Biomechanical Engineering 114: 427–434.Google Scholar
  21. Turing, A.M. (1952). The Chemical Basis of Morphogenesis. Philosophical Transactions of the Royal Society, London B237: 37–72.Google Scholar
  22. Zahalak, G.I. (1997). Can Muscle Fibers Be Stable on the Descending Limbs of Their Sarcomere Length-Tension Relations? Journal of Biomechanics 30: 1179–1182.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Larry A. Taber
    • 1
  1. 1.Department of Biomedical EngineeringWashington UniversitySt. LouisUSA. Tel

Personalised recommendations